code: def lie_heuristic_abaco1_product(match, comp=False):
      r"""
      The second heuristic uses the following two assumptions on `\xi` and `\eta`
  
      .. math:: \eta = 0, \xi = f(x)*g(y)
  
      .. math:: \eta = f(x)*g(y), \xi = 0
  
      The first assumption of this heuristic holds good if
      `\frac{1}{h^{2}}\frac{\partial^2}{\partial x \partial y}\log(h)` is
      separable in `x` and `y`, then the separated factors containing `x`
      is `f(x)`, and `g(y)` is obtained by
  
      .. math:: e^{\int f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)\,dy}
  
      provided `f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)` is a function
      of `y` only.
  
      The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as
      `\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first assumption
      satisfies. After obtaining `f(x)` and `g(y)`, the coordinates are again
      interchanged, to get `\eta` as `f(x)*g(y)`
  
  
      References
      ==========
      - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order
        ODE Patterns, pp. 7 - pp. 8
  
      """
  
      xieta = []
      y = match["y"]
      h = match["h"]
      hinv = match["hinv"]
      func = match["func"]
      x = func.args[0]
      xi = Function("xi")(x, func)
      eta = Function("eta")(x, func)
  
      inf = separatevars(
          ((log(h).diff(y)).diff(x)) / h**2, dict=True, symbols=[x, y]
      )
      if inf and inf["coeff"]:
          fx = inf[x]
          gy = simplify(fx * ((1 / (fx * h)).diff(x)))
          gysyms = gy.free_symbols
          if x not in gysyms:
              gy = exp(integrate(gy, y))
              inf = {eta: S.Zero, xi: (fx * gy).subs(y, func)}
              if not comp:
                  return [inf]
              if comp and inf not in xieta:
                  xieta.append(inf)
  
      u1 = Dummy("u1")
      inf = separatevars(
          ((log(hinv).diff(y)).diff(x)) / hinv**2,
          dict=True,
          symbols=[x, y],
      )
      if inf and inf["coeff"]:
          fx = inf[x]
          gy = simplify(fx * ((1 / (fx * hinv)).diff(x)))
          gysyms = gy.free_symbols
          if x not in gysyms:
              gy = exp(integrate(gy, y))
              etaval = fx * gy
              etaval = (etaval.subs([(x, u1), (y, x)])).subs(u1, y)
              inf = {eta: etaval.subs(y, func), xi: S.Zero}
              if not comp:
                  return [inf]
              if comp and inf not in xieta:
                  xieta.append(inf)
  
      if xieta:
          return xieta
  
imports: from itertools import islice
  from sympy.core import Add, S, Mul, Pow
  from sympy.core.exprtools import factor_terms
  from sympy.core.function import Function, AppliedUndef, expand
  from sympy.core.relational import Equality, Eq
  from sympy.core.symbol import Symbol, Wild, Dummy, symbols
  from sympy.functions import exp, log
  from sympy.integrals.integrals import integrate
  from sympy.polys import Poly
  from sympy.polys.polytools import cancel, div
  from sympy.simplify import collect, powsimp, separatevars, simplify
  from sympy.solvers import solve
  from sympy.solvers.pde import pdsolve
  from sympy.utilities import numbered_symbols
  from sympy.solvers.deutils import _preprocess, ode_order
  from .ode import checkinfsol
label_desc: Use image node_id 5 for calling a global function with example usage: lie_heuristic_abaco1_product(match, comp) and returns: xieta, list, list